Are the maps I create only available to myself or people I designate ? I am familiar with google maps but not this service. Google maps consistently eliminates map off their site. Does this happen here as well ?

OSM is not a service to create personal maps. See the likes of umap and mapbbshare (which are osm-based) for that.

OSM is a project to gather the actual map data. Displaying rendered maps on osm.org is in a sense just a nice bonus. We all work on the same map. If you contribute a map changeset or a note, it is shared with everybody.